Zermelo Navigation

This is a worked example of the sort of optimization problem that
is typical of the calculus of variations. This document contains
the text of the lecture presentation on this example, and you
may follow along the transcript below, if you like.

Contents

Instructor: Now I'd like to discuss a particular
case of interest in this sort of optimization problem. It's called
the Zermelo Navigation Problem, discussed by E. Zermelo in
1931.

The physical situation to be modeled is the following one.
We have a boat capable of a certain maximum speed and we want
to understand how to navigate on a body of water, choosing paths
that go from an originating point O to a destination D in the
least possible time.

Now, if there is no current or wind to consider, so that it
is equally easy to go in any direction, we can just follow the
straight line segment from O to D. (We're assuming that
the body of water is convex, otherwise, we'll have to take the
shape of the coastline into account. This can be done, but I
won't do this today, so that we can focus on a different aspect
of the problem.)

Now suppose, instead, that there is a current, i.e, that the
water is moving. Now, when you point the boat in a particular
direction, the actual velocity vector of the boat with respect to
the fixed bottom is the sum of the heading velocity and the
current velocity. In a steady current, for example, this causes the
boat to be dragged sideways.

Direct heading in a steady stream

You might think, at first glance, that the right thing to do
is just keep heading directly for D. Such a strategy will
result in the black paths in a steady current (drawn in blue), whereas
aiming steadily at the appropriate fixed angle to the current will result
in the green paths, i.e., straight lines.

It's not hard to see that the green paths have a shorter time
traverse. Of course,
since we are assuming a steady current, this problem is invariant under
translation. Thus, our earlier Theorem on
shortest paths with translation symmetry shows that the straight
lines must be the shortest paths for time traversal. Actually, it's an
interesting calculus exercise to determine the traverse time of the black
paths to see this directly. I'll leave this to you. If you want hints,
consult the hints page for today's lecture.
After you've done this, you might want to think about why the 'head
directly for the goal' strategy might still have some good points, even
though it's not optimal. After all, there must be some reason that
it seems to occur to most people first!